{"id":2997,"date":"2019-06-28T14:45:32","date_gmt":"2019-06-28T14:45:32","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=2997"},"modified":"2022-07-19T14:37:58","modified_gmt":"2022-07-19T14:37:58","slug":"apply-transformations","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/univariate-random-variables\/apply-transformations\/","title":{"rendered":"Apply Transformations"},"content":{"rendered":"<p>Transformations allow us to find the distribution of a function of random variables. There are different methods of applying transformations.<\/p>\r\n<h2><strong>The Method of Distribution Function<\/strong><\/h2>\r\n<p>Given a random variable \\(Y\\) that is a function of a random variable \\(X\\), that is\u00a0 \\(Y=u(X)\\), we can find the cumulative density function of \\(Y\\), that is \\(G(y)\\), directly and then differentiate this cumulative distribution function (CDF) to get the PDF of \\(Y\\), that is \\(g(y)\\):<\/p>\r\n<p>$$ \\begin{align*} G\\left(y\\right)&amp;=P\\left(Y\\le y\\right)=P[u\\left(X\\right)\\le y] \\\\ g\\left(y\\right)&amp;=G^\\prime\\left(y\\right) \\end{align*} $$<\/p>\r\n<h4>Example: The Method of Distribution Function<\/h4>\r\n<p>A random variable \\(X\\) has a uniform distribution over the interval \\([-2,2]\\), and another variable \\(Y\\) is defined as \\(Y=X^2\\).<\/p>\r\n<p>Find the probability density function of \\(Y\\), \\(f(y)\\).<\/p>\r\n<p><strong>Solution<\/strong><\/p>\r\n<p>We first find the cumulative density function of \\(Y\\), that is \\(G(y)\\):<\/p>\r\n<p>$$\\begin{align}G(y)&amp;amp;=P(Y\\leq y)\\\\ &amp;amp; =P(X^2\\leq y)\\\\ &amp;amp;=P(X\\leq \\sqrt{y})\\\\ &amp;amp;=P(-\\sqrt {y}\\le X\\le \\sqrt {y}\\\\ &amp;amp;=\\int_{-\\sqrt y}^{\\sqrt y}{f\\left(x\\right)dx=2*\\int_{0}^{\\sqrt y}{\\frac{1}{4}dx=2*\\frac{1}{4}\\left[x\\right]_0^{\\sqrt y}}}\\\\ &amp;amp;=\\frac{\\sqrt y}{2}\\end{align}$$<\/p>\r\n<p>Thus,<\/p>\r\n<p>$$g\\left(y\\right)=\\frac{d}{dy}\\left(\\sqrt y\\right)=\\frac{1}{4\\sqrt y}\\ ,\\ for-2\\le y\\le 2$$<\/p>\r\n<h3><strong>The Method of Transformation\u00a0<\/strong><\/h3>\r\n<p>In this method, we consider a variable \\(X\\), whose PDF, \\(f(x)\\), is given and another variable \\(Y\\), such that, \\(Y=u(X) \\) where \\(u(x)\\) is either an increasing or decreasing function of \\(x\\) that is \\(u\\left(x\\right) &gt; 0\\).<\/p>\r\n<p>We first find the inverse function of\u00a0 \\(u\\left(x\\right)\\), that is \\(x=u^{-1}\\left(y\\right)\\).<\/p>\r\n<p>We then evaluate:<\/p>\r\n<p>$$\\frac{du^{-1}}{dy}=\\frac{d\\left[u^{-1}\\left(y\\right)\\right]}{dy}$$<\/p>\r\n<p>We then finally find \\(f(y)\\), using the formula:<\/p>\r\n<p>$$f_Y\\left(y\\right)=f_X\\left[u^{-1}\\left(y\\right)\\right]\\left|\\frac{du^{-1}}{dy}\\right|$$<\/p>\r\n<h4>Example: Applying Transformation \u2013 Use of the Inverse Function<\/h4>\r\n<p>Let \\(X\\) be a random variable with a PDF:<\/p>\r\n<p>$$f(x)=\\begin{cases} x^2 +\\frac{2}{3},&amp;0 &lt; x &lt; 1\\\\ 0, &amp; \\text{otherwise}\\end{cases}$$<\/p>\r\n<p>Find the PDF of \\(U=2x-3\\).<\/p>\r\n<p><strong>Solution<\/strong><\/p>\r\n<p>Let \\(z(X)=2x-3\\).<\/p>\r\n<p>If \\(u=2x-3\\); and making \\(x\\) the subject of the formula:<\/p>\r\n<p>$$x=z^{-1}(u)=\\frac{u+3}{2}$$<\/p>\r\n<p>And thus,<\/p>\r\n<p>$$\\frac{{dz}^{-1}(u)}{du}=\\frac{dx}{du}=\\frac{1}{2}$$<\/p>\r\n<p>Now,<\/p>\r\n<p>$$\\begin{align}f\\left(u\\right)&amp;=f_X\\left(z^{-1}\\left(u\\right)\\right)\\left|\\frac{dx}{du}\\right|\\\\&amp;=\\begin{cases} [(u+3\/2)^2+2\/3]|1\/2|=1\/2 (u+3\/2)^2+1\/3, &amp;-3 &lt; u &lt; 1\\\\0, &amp; \\text{elsewhere}\\end{cases} \\end{align}$$<\/p>\r\n<h3><strong>Change of Variable Technique<\/strong><\/h3>\r\n<p>We can use the <strong>change-of-variable technique<\/strong> to\u00a0 find the PDF of a transformed variable, \\(Y\\).<\/p>\r\n<p>$$ g\\left( y \\right) =f\\left[ \\upsilon \\left( y \\right) \\right] |{ \\upsilon }^{ \\prime }\\left( y \\right) | $$<\/p>\r\n<p>where \\(\\upsilon \\left( y \\right)\\) is the inverse function of \\(y\\).<\/p>\r\n<h4>Example: Change-of-variable Technique<\/h4>\r\n<p>Given the following probability density function of a continuous random variable:<\/p>\r\n<p>$$ f\\left( x \\right) =\\begin{cases} { x }^{ 2 }+\\cfrac { 2 }{ 3, } &amp; 0&lt; x &lt; 1 \\\\ 0, &amp; \\text{otherwise} \\end{cases} $$<\/p>\r\n<p>Let \\(Y = X \u2013 50\\).<\/p>\r\n<p>Find \\(g(y)\\).<\/p>\r\n<p><strong>Solution<\/strong><\/p>\r\n<p>\\(v\\left(y \\right) = Y + 50\\)<\/p>\r\n<p>\\(v^{\\prime} \\left(y \\right) = 1\\)<\/p>\r\n<p>$$ g\\left( y \\right) =f\\left[ \\upsilon \\left( y \\right) \\right] \\left[ { \\upsilon }^{ \\prime }\\left( y \\right) \\right] =\\left[ { \\left( Y+50 \\right) }^{ 2 }+\\frac { 2 }{ 3 } \\right] \\left[ 1 \\right] ={ \\left( Y+50 \\right) }^{ 2 }+\\frac { 2 }{ 3 } $$<\/p>\r\n<p>The PDF of a <strong>discrete transformed random variable<\/strong> can be found in a similar manner, as shown in the formula below:<\/p>\r\n<p>$$ g\\left( y \\right) =f\\left[ \\upsilon \\left( y \\right) \\right] $$<\/p>\r\n<p><strong>Note:<\/strong> \\(|{ \\upsilon }^{ \\prime }\\left( y \\right) |\\) is not needed in this case.<\/p>\r\n<p><em><strong>Learning Outcome:<\/strong><\/em><\/p>\r\n<p><em><strong>Topic 2g: Univariate Random Variables &#8211; Apply transformations.<\/strong><\/em><\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>Transformations allow us to find the distribution of a function of random variables. There are different methods of applying transformations. The Method of Distribution Function Given a random variable \\(Y\\) that is a function of a random variable \\(X\\), that&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[98],"tags":[],"class_list":["post-2997","post","type-post","status-publish","format-standard","hentry","category-univariate-random-variables","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Apply Transformations - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/univariate-random-variables\/apply-transformations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Apply Transformations - CFA, FRM, and Actuarial Exams Study Notes\" \/>\n<meta property=\"og:description\" content=\"Transformations allow us to find the distribution of a function of random variables. 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